Showing papers on "Extended finite element method published in 1986"

Random field finite elements

Abstract: The probabilistic finite element method (PFEM) is formulated for linear and non-linear continua with inhomogeneous random fields. Analogous to the discretization of the displacement field in finite element methods, the random field is also discretized. The formulation is simplified by transforming the correlated variables to a set of uncorrelated variables through an eigenvalue orthogonalization. Furthermore, it is shown that a reduced set of the uncorrelated variables is sufficient for the second-moment analysis. Based on the linear formulation of the PFEM, the method is then extended to transient analysis in non-linear continua. The accuracy and efficiency of the method is demonstrated by application to a one-dimensional, elastic/plastic wave propagation problem and a two-dimensional plane-stress beam bending problem. The moments calculated compare favourably with those obtained by Monte Carlo simulation. Also, the procedure is amenable to implementation in deterministic FEM based computer programs.

A new finite element formulation for computational fluid dynamics: III. The generalized streamline operator for multidimensional advective-diffusive systems

Abstract: A finite element method based on the SUPG concept is presented for multidimensional advective-diffusive systems. Error estimates for the linear case are established which are valid over the full range of advective-diffusive phenomena.

The $h-p$ version of the finite element method with quasiuniform meshes

Abstract: : The classical error estimates for the h-version of the finite element method are extended for the h-p version. The estimates are expressed as explicit functions of h and p are shown to be optimal. The estimates are given for the case where the solution u (H sub k and the case when u has singularities at the corners of the domain. (Author)

Dynamics of viscoelastic structures: a time-domain finite element formulation

Abstract: Mathematical models of elastic structures have become very sophisticated: given the crucial material properties (mass density and the several elastic moduli), computer-based techniques can be used to construct exotic finite element models. By contrast, the modeling of damping is usually very primitive, often consisting of no more than mere guesses at “modal damping factors.” The aim of this paper is to raise the modeling of viscoelastic structures to a level consistent with the modeling of elastic structures. Appropriate material properties are identified which permit the standard finite element formulations used for undamped structures to be extended to viscoelastic structures. Through the use of “dissipation” coordinates, the canonical “M , K ” form of the undamped motion equations is expanded to encompass viscoelastic damping. With this formulation finite element analysis can be used to model viscoelastic damping accurately.

The h-p version of the finite element method

Abstract: The paper is the first of the series of two which analyses the h-p version of the finite element method in two dimensions. It proves the basic approximation results which in part 2 will be generalized and applied in a computational setting. The main result is that the h-p version leads to an exponential rate of convergence when solving problems with piecewise analytic data.

Comparison of finite element techniques for solidification problems

Abstract: The accuracies of the computed temperatures of a liquid in a corner region under freezing conditions are compared for various fixed-grid finite element techniques using the analytical solution for this problem as a reference. In the finite element formulation of the problem different time-stepping schemes are compared: the implicit Euler-backward algorithm combined with an iterative scheme and two three-time-level methods—the Lees algorithm and a Dupont algorithm, which are both applied as non-iterative schemes. Furthermore, different methods for handling the evolution of latent heat are examined: an approximation method suggested by Lemmon and one suggested by Del Giudice, both using the enthalpy formulation as well as a fictitious heat-flow method presented by Rolph and Bathe. Results of calculations performed with the consistent heat-capacity matrix are compared with those performed with a lumped heat-capacity matrix.

An arbitrary Lagrangian-Eulerian finite element method for path-dependent materials

Abstract: The conservation laws, the constitutive equations, and the equation of state for path-dependent materials are formulated for an arbitrary Lagrangian-Eulerian finite element method. Both the geometrical and material nonlinearities are included in this setting. Computer implementations are presented and an elastic-plastic wave propagation problem is used to examine some features of the proposed method.

The h , p and h-p versions of the finite element methods in 1 dimension . Part III. The adaptive h-p version.

Abstract: : The paper is the third and final part in the series of three devoted to the detailed analysis of the three basic versions of the finite element method in one dimension. The first part analyzed the p-version, and the second part concentrated on the h and h-p version. This paper analyzes a theoretical frame of the adaptive h-p version and based on it the authors provide concrete algorithm for the one dimensional problem. It is proven that in the case that the solution has x sub alpha-type singularity, the adaptive algorithm give an exponential rate of convergence, very close to the optimal one analyzed in the second part of the paper. Additional keyword: Error analysis.

The hybrid-Trefftz finite element model and its application to plate bending

Abstract: This paper presents a new hybrid element approach and applies it to plate bending. In contrast to more conventional models, the formulation is based on displacement fields which fulfil a priori the non-homogeneous Lagrange equation (Trefftz method). The interelement continuity is enforced by using a stationary principle together with an independent interelement displacement. The final unknowns are the nodal displacements and the elements may be implemented without any difficulty in finite element libraries of standard finite element programs. The formulation only calls for integration along the element boundaries which enables arbitrary polygonal or even curve-sided elements to be generated. Where relevant, known local solutions in the vicinity of a singularity or stress concentration may be used as an optional expansion basis to obtain, for example, particular singular corner elements, elements presenting circular holes, etc. Thus a high degree of accuracy may be achieved without a troublesome mesh refinement. Another important advantage of the formulation is the possibility of generating by a single element subroutine a large number of various elements (triangles, quadrilaterals, etc.), presenting an increasing degree of accuracy. The paper summarizes the results of numerical studies and shows the excellent accuracy and efficiency of the new elements. The conclusions present some ideas concerning the adaptive version of the new elements, extension to nonlinear problems and some other developments.

Mesh design for the p-version of the finite element method

Abstract: When properly designed meshes are used then the performance of the p-extension is very close to the best performance attainable by the finite element method. Proper mesh design depends on the exact solution, however. Because the exact solution is not known a priori, initial mesh design is generally based on certain assumptions concerning the exact solution which must be tested in the post-solution phase to ensure reliability and accuracy of data computed from the finite element solution. In this paper general guidelines are presented for prior design of meshes, and procedures for post-solution testing are described and illustrated by examples.

On the construction of optimal mixed finite element methods for the linear elasticity problem

Abstract: The mixed finite element method for the linear elasticity problem is considered. We propose a systematic way of designing methods with optimal convergence rates for both the stress tensor and the displacement. The ideas are applied in some examples.

Resultant-stress degenerated-shell element

Abstract: In this paper, a resultant-stress degenerated-shell element is described and a variety of numerical examples, including the post-buckling analysis of an axially loaded perfect cylinder, are presented. The general degenerated nonlinear shell theory of Hughes and Liu is employed in deriving this resultant-stress degenerated-shell element. Contrary to the traditional integration through the thickness approach, which assumes no coupling between the in-plane and transverse material and structural response matrices, the present approach can permit use of arbitrary, three-dimensional (3-D) nonlinear constitutive equations. Furthermore, explicit expressions of the element matrices for a 4-node shell element are developed. This rank-sufficient 4-node shell element, termed the resultant-stress degenerated-shell (RSDS) element, avoids the need for the costly numerical quadrature function evaluations of the element matrices and force vectors. And thus there are large increases in computational efficiency with this method. The comparisons of this RSDS element with six other shell elements are also given in this paper.

Finite Element Methods in Mechanics

Abstract: This is a textbook written for mechanical engineering students at first-year graduate level. As such, it emphasizes the development of finite element methods used in applied mechanics. The book starts with fundamental formulations of heat conduction and linear elasticity and derives the weak form (i.e. the principle of virtual work in elasticity) from a boundary value problem that represents the mechanical behaviour of solids and fluids. Finite element approximations are then derived from this weak form. The book contains many useful exercises and the author appropriately provides the student with computer programs in both BASIC and FORTRAN for solving them. Furthermore, a workbook is available with additional computer listings, and also an accompanying disc that contains the BASIC programs for use on IBM-PC microcomputers and their compatibles. Thus the usefulness and versatility of this text is enhanced by the student's ability to practise problem solving on accessible microcomputers.

Element-by-element linear and nonlinear solution schemes

Abstract: The preconditionded conjugate gradient method for solving a linear algebraic system of equations is recast to a form that permits sequential element-by-element calculations suitable for computations with finite element methods. This strategy has been implemented for solving the linear systems arising in a finite element approximation for the standard test example of Laplace's equation. The element-by-element strategy has also been applied to the sequence of linear systems obtained using a successive approximation scheme for a representative class of nonlinear problems. Little storage is needed for these schemes, and test computations have been made on microprocessor, minicomputer, main-frame computers and special processors. The approach also appears appealing for calculations on parallel processors since individual element computations can be done in parallel.

Numerical Modeling of Immiscible Organic Transport at the Hyde Park Landfill

Abstract: In this paper, a two-dimensional two-phase mathematical model based on Darcy's law and conservation of mass for each liquid is presented. The numerical model is based on a generalized method of weighted residuals in conjunction with the finite element method and linear quadrilateral isoparametric elements. To alleviate numerical problems associated with hyperbolic equations, upstream weighting of the spatial terms in the model has been incorporated. The theoretical and numerical accuracy of the model is verified by comparison of simulation results with those from an existing one-dimensional two-phase flow simulator. The finite element model is used to simulate the migration of an immiscible organic solvent in groundwater, from a chemical waste disposal site located north of Niagara Falls, New York. The effects of uncertainty regarding porous media heterogeneities and anisotropy are examined, and it is concluded that the extent of immiscible contaminant migration is greatly sensitive to these parameters.

Finite Element Methods for Nonlinear Problems

Abstract: The first € price and the £ and $ price are net prices, subject to local VAT. Prices indicated with * include VAT for books; the €(D) includes 7% for Germany, the €(A) includes 10% for Austria. Prices indicated with ** include VAT for electronic products; 19% for Germany, 20% for Austria. All prices exclusive of carriage charges. Prices and other details are subject to change without notice. All errors and omissions excepted. P.G. Bergan, K.-J. Bathe, W. Wunderlich (Eds.) Finite Element Methods for Nonlinear Problems

Adaptive grid-design methods for finite delement analysis

Abstract: This paper is concerned with an introduction of a concept of adaptive grid design for finite element analysis by combining numerical grid-generation methods and adaptive finite element methods. Development of a finite model is considered as a design problem similar to structural optimization problems.

A new method for the coupling of finite element and boundary element discretized subdomains of elastic bodies

Abstract: A new and efficient approach for the coupling of subregions of elastic solids discretized by means of finite elements (FE) and boundary elements (BE), respectively, is presented. The method is characterized by so-called ‘bi-condensation’ of nodal degrees of freedom followed by the transformation of the resulting BEM-related traction-displacement equations for the interface(s) of the BE subregion(s) and the FE subdomain(s) to ‘FEM-like’ force-displacement relations which are assembled with the FEM-related force-displacement equations for the interface(s). The presented ‘local FE coupling approach’ is computationally more economic than a global coupling approach since it only requires the inversion of BEM-related coefficient matrices referred to the interfaces of BE subregions and FE subdomains. Depending on whether the principle of virtual displacements or the principle of minimum of potential energy is used for the generation of force-displacement equations for the coupling interface(s), unsymmetric or symmetric coefficient matrices are obtained. Since the two principles are mechanically equivalent, identical results would be achieved in the limit of finite discretizations. The numerical investigation has shown that, depending on the problem and the discretization, the results obtained on the basis of symmetric coefficient matrices may be poor. This applies to ‘edge problems’ characterized by discontinuous tractions along the edges. On the basis of unsymmetric coefficient matrices, however, satisfactory results are obtained even for relatively coarse discretizations.

A mesh re-zoning technique for finite element simulations of metal forming processes

Abstract: Based on some fundamental properties of finite element approximations, a mesh re-zoning scheme is proposed for finite element simulations of metal forming problems. It is demonstrated that this technique is indispensable in analyzing many difficult forming processes, especially when there exist corners or very irregular shapes on the boundaries. The algorithm is tested by a backward extrusion process and direct extrusion through a square die.

A temperature‐based finite element solution for phase‐change problems

Abstract: A finite element procedure for solving multidimensional phase change problems is described. The algorithm combines a temperature formulation with a finite element treatment of the differential equation and discontinuous integration within the two-phase elements to avoid the necessity of regularization. A new criterion for the computation of the iteration matrix is proposed. It is based on a quasi-Newton correction of the Jacobian matrix for conduction problems without change of phase. A set of test problems with exact solution is analysed and demonstrates that the procedure can accurately evaluate the front position and temperature history with a reasonable computational effort.